What really is a normal mode? Maybe it's because of my teachers but I find it really abstract. I know that "numerically" corresponds to the eigenvectors of the equation $\ddot{X}= -M^{-1}KX$ with the solution of the form $Ae^{-iwt}$ already plugged in, but what is the physical meaning of those eigenvectors in the first place? Is it the amplitude of the system when in normal mode? And (there you go) what does it mean to be in a normal mode?
I appreciate your answer in advance.
In a normal mode, all parts of the system are oscillating with one single frequency. If we write the positions of the objects as a column vector $X$, the (linearized undamped undriven) coupled differential equations can, in general, be written as $$\ddot{X} = MX$$ where $M$ represents the coefficients. To solve for the normal modes, we need to assume that the frequency is $\omega$, so we substitute $X = Ae^{i\omega t}$ to get $$-\omega^2 A = MA$$ which is the eigenvalue problem that we were looking for. Therefore, we can see that the eigenvalues $-\omega^2$ represent the pure frequencies of the normal modes, while the eigenvectors $A$ represent the relative amplitudes of each object.
This is entirely analogous to the ordinary differential equation $\ddot{x} = -\omega^2 x$ for a single oscillator. The general solution is then a linear combination of the normal modes.
Imagine a chain made of a line of springs with some mass $m$ located between each spring and the next.
Now suppose you wobble this system in some complicated wobbly motion (not a normal mode). All the masses will move up and down by different amounts and in some complicated motion. If you could hear the sound waves produced by the motion then it would sound like noise.
Now suppose instead that you carefully arrange that one of the masses goes up and down in a strictly periodic way: just one frequency, just one 'note' if you could hear it. The adjacent masses will pick up this motion too, and then the ones adjacent to those, etc. The motion throughout the system will stay complicated for a while, but eventually it may settle into a pattern where all the masses are going up and down at the same frequency. They don't have to be in phase: some may be going up while others go down, but this phase difference will be constant in time. Also they don't have to have the same amplitude: some may go up and down more than others. If you looked at the system it would like as if it had a standing wave, with all the motions having exactly the same frequency, repeating over and over.
This sort of motion is called a 'mode of oscillation' or just 'mode' for short. If, in the absence of damping, you do not need to provide an external force because the system just carries on oscillating at a single frequency throughout, then it is called a 'normal mode'.
For any given set of springs and masses, the normal mode motion can only take place at one of a discrete set of frequencies (this is what the eigenvalues tell you). And each normal mode has its own characteristic shape of displacements along the system (this is what the eigenvectors tell you).
I talked about springs and masses, but the concept is more general, and has quite a widespread application in physics, pretty much whenever something is more complicated than a single particle and can undergo some sort of oscillatory motion.
After passing by this question of mine recently, I thought I should give it an answer as I'm now able to do it.
Normal modes is a concept quite beautiful - @Andrew Steane almost answered it for me actually - but basically you have this super complicated movement, or it could be a wave, but the idea is it is something really complex and complicated, and the premise is you can actually recreate this movement using overlays of really simple movements.
In fact, if we're talking for example in a 1-D coupled oscillator, which can generate pretty complex movements to analyse, you can break it down in actually two simple movements, and the complex movement will be a sort of overlap of these two simple movements. These two simple movements are called normal modes.
The idea is exactly the same with vectors, you can represent any vector, no matter how complicated it is, with the basis vectors. Normal modes work as basis vectors (in fact, if they're correctly normalized, they're the same thing) - you can sum them up to recreate the super complicated movement that is happening.
The visualization of adding simple movements that ultimately gives the actual movement is not something easy to imagine, but to convince me that movements can actually be summed up I like to think in two simple movements, for example a 1-D oscillator vibrating with a certain frequency (in a certain direction, sort of speak) and the other simple movement would be the same 1-D oscillator vibrating with the same frequency but with a difference in phase of $\pi$ (in other words, it's vibrating in the opposite direction of the first movement, sort of speak). It makes sense that if we add these two simple movements (with equal amount of contribution of each movement), the overall effect would be that the 1-D oscillator would be in rest, as the simple movements cancel each other out. So it makes sense that we can generate other movements summing other movements.
I hope my answer could help you understand the intuition behind normal modes, something back then it was something hard for me to grasp.